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2. the sum of zero terms (the empty sum) is 0; the product of zero terms (the empty product) is 1

3. the degree of the zero polynomial is -∞

4. the maximum of zero numbers is -∞; the minimum of zero numbers is +∞

5. the determinant of the 0x0 matrix (the empty matrix, with no entries) is 1

6. the empty graph is disconnected, and so is the empty topological space, for the same reason that 1 isn't a prime number

7. there's a different empty manifold in each dimension: a zero-dimensional empty manifold, a 1-dimensional empty manifold, etc. (and of course none of them are connected or path-connected)

8. the zero-dimensional vector space (consisting only of the vector 0) has a unique basis, which is empty

Remember when you were in school, and your mathematics teacher told you that 00 = 1, and you got very confused? This is the type theory equivalent of that. So what’s this all about? It’…

9. (this one is new to me, thanks

@achambertloir

!) the zero endomorphism of the zero banach space has operator norm 0; the endomorphisms of the zero banach space do still form a unital banach algebra but only if you relax the unit axiom to |id| \le 1

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Antoine Chambert-Loir

@achambertloir

· Aug 12, 2020

Replying to @achambertloir and @QiaochuYuan

By the way, for the operator norm of an endomorphism, the supremum is taken in R_+ adjoined +infinity, so that the norm of the identity endo of the null space is 0.

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I wrote a bit about this

semantic.org

Empty and Unit Types